Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2833 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C3×#400 | C2 |
| hypercenter | #400 | D3 |
| hypocenter | C3 | #2833 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #400 | |
332 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(6,10)(8,11)(20,24,25,22)(21,26,28,29)(23,30,33,32)(27,35,31,34);(2,3)(5,9)(8,11)(14,18)(16,19)(20,23,25,33)(21,27,28,31)(22,30,24,32)(26,34,29,35);(2,3)(4,6)(5,8)(7,10)(9,11)(12,14)(13,16)(15,18)(17,19)(20,22,25,24)(21,26,28,29)(23,32,33,30)(27,35,31,34);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,15,17)(14,16,18,19)(20,21)(22,29)(23,31)(24,26)(25,28)(27,33)(30,34)(32,35)Polycyclic presentation:
2^7,3|a^2=e,b^2=eg,c^2=eg,d^2=eg,b^a=beg,c^a=cf,h^a=h^2,c^b=cg,d^b=defg,h^b=h^2,d^c=deg,h^c=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10707897, #10707901, #10708381, … |
| maximal normal subgroup of | #10707897, #10707901, #10708381, … |
| maximal quotient of | #10725887, #10725973, #10726006, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10899);